Jeroen Elassaiss-Schaap (1,2), Kevin Duisters (3)
(1) PD-value, the Netherlands, (2) LACDR, Leiden University, the Netherlands, (3) Mathematical Institute, Leiden University, the Netherlands
Introduction: Pharmacometric models with lognormal variability have become commonplace in pharmacokinetic-pharmacodynamic investigations. Standard approaches in interpretation and reporting of parameters with lognormal distribution however are mostly as if these parameters were specified as normal, and with the coefficient of variation (CV) as the most popular statistic. Unfortunately, the CV captures the behavior of the lognormal distribution poorly. Pitfalls in interpretation and reporting include the location of prediction intervals, differences between median and mean, and sharpening of the peak of the distribution with increasing variability.
Objectives:
- Determine the extent to which the lognormal distribution can be safely interpreted as a normal distribution
- Define additional metrics to support interpretation and avoid some common pitfalls.
Methods: The properties of the lognormal distribution are derived mathematically from first principles to ensure correct implementation. The properties of the lognormal distribution as it depends on the magnitude of variability are subsequently explored graphically. The extent to which the lognormal can be interpreted as the normal distribution is evaluated using these properties. Within this evaluation, approximation of the lognormal distribution by the normal distribution is performed using formal optimization of the mean and standard deviation through the Kullback-Leibner divergence [1], but also using a rule of thumb. The positions of the approximated distributions at the 10th and the 90th percentiles are compared to the true position, and the relative error is discussed.
Results: The lognormal distribution is rather similar to the normal distribution at low values for the lognormal variability (omega), with CV values << 0.1. With increasing omega, the distribution becomes more asymmetric and flatter up to an omega of 1, CV=131%, where the peak of the density function is lowest. With even higher omega, a less widely known feature of the lognormal distribution manifests itself, and peak density increases again but occurs at values close to zero. At an omega of 2, the lognormal distribution is unequivocally different from the normal distribution with a sharp peak close to zero and a heavy tail towards higher values, and any attempt to interpret it as normal fails.
To establish the limits at which such interpretation fails, normal-approximated percentiles were compared to true lognormal percentiles. The error exceeded 10% of true values at an omega of 0.25 (CV=25.4%) for formal approximations of the 10th percentile. Approximations for the 90th percentile exceeded the 10-% mark at an omega of 1, or even 1.1 in case of the rule-of-thumb method (CV=153%).
To develop tools to support proper interpretation of the lognormal distribution at higher values, new metrics were selected to be reasonably intuitive, capture the heavy tails, asymmetry and peak-sharpening features of lognormal variability and not require extensive calculations. The asymmetry and also the tailing are represented by the median-to-mean ratio (MMR) while the peak-sharpening is represented by the mode density inflation (MDI), where the peak density at the mode is normalized to its minimal value.
Conclusion: The properties of the lognormal distribution were explored graphically and mathematically, and boundaries were established on the extent to which the lognormal distribution can be interpreted as normal: up to a lognormal variability (omega) of 0.25 for typical use cases. Two metrics are presented that have the potential to intuitively capture the shape of lognormal variability: the MMR and MDI. A framework to define how the MMR and MDI could support proper interpretation of the lognormal distribution is provided.
References:
[1] Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Statist 22(1):79–86
Reference: PAGE () Abstr 9276 [www.page-meeting.org/?abstract=9276]
Poster: Oral: Other Topics